The Peter-Weyl theorem, and non-abelian Fourier analysis on compact groups

Let be a compact group. (Throughout this post, all topological groups are assumed to be Hausdorff.) Then has a number of unitary representations, i.e. continuous homomorphisms to the group of unitary operators on a Hilbert space , equipped with the strong operator topology. In particular, one has the left-regular representation, where we equip with its normalised Haar measure (and the Borel -algebra) to form the Hilbert space , and is the translation operation

We call two unitary representations and isomorphic if one has for some unitary transformation , in which case we write .

Given two unitary representations and , one can form their direct sum in the obvious manner: . Conversely, if a unitary representation has a closed invariant subspace of (thus for all ), then the orthogonal complement is also invariant, leading to a decomposition of into the subrepresentations, . Accordingly, we will call a unitary representation irreducible if is nontrivial (i.e. ) and there are no nontrivial invariant subspaces (i.e. no invariant subspaces other than and ); the irreducible representations play a role in the subject analogous to those of prime numbers in multiplicative number theory. By the principle of infinite descent, every finite-dimensional unitary representation is then expressible (perhaps non-uniquely) as the direct sum of irreducible representations.

The Peter-Weyl theorem asserts, among other things, that the same claim is true for the regular representation:

Theorem 1 (Peter-Weyl theorem) Let be a compact group. Then the regular representation is isomorphic to the direct sum of irreducible representations. In fact, one has , where is an enumeration of the irreducible finite-dimensional unitary representations of (up to isomorphism). (It is not difficult to see that such an enumeration exists.)

In the case when is abelian, the Peter-Weyl theorem is a consequence of the Plancherel theorem; in that case, the irreducible representations are all one dimensional, and are thus indexed by the space of characters (i.e. continuous homomorphisms into the unit circle ), known as the Pontryagin dual of . (See for instance my lecture notes on the Fourier transform.) Conversely, the Peter-Weyl theorem can be used to deduce the Plancherel theorem for compact groups, as well as other basic results in Fourier analysis on these groups, such as the Fourier inversion formula.

Because the regular representation is faithful (i.e. injective), a corollary of the Peter-Weyl theorem (and a classical theorem of Cartan) is that every compact group can be expressed as the inverse limit of Lie groups, leading to a solution to Hilbert’s fifth problem in the compact case. Furthermore, the compact case is then an important building block in the more general theory surrounding Hilbert’s fifth problem, and in particular a result of Yamabe that any locally compact group contains an open subgroup that is the inverse limit of Lie groups.

Because of the above motivation, I have decided to write some notes on how the Peter-Weyl theorem is proven. This is utterly standard stuff in abstract harmonic analysis; these notes are primarily for my own benefit, but perhaps they may be of interest to some readers also.

— 1. Proof of the Peter-Weyl theorem —

Throughout these notes, is a fixed compact group.

Let and be unitary representations. An (linear) equivariant map is defined to be a continuous linear transformation such that for all .

A fundamental fact in representation theory, known as Schur’s lemma, asserts (roughly speaking) that equivariant maps cannot mix irreducible representations together unless they are isomorphic. More precisely:

Lemma 2 (Schur’s lemma for unitary representations) Suppose that and are irreducible unitary representations, and let be an equivariant map. Then is either the zero transformation, or a constant multiple of an isomorphism. In particular, if , then there are no non-trivial equivariant maps between and .

Proof: The adjoint map of the equivariant map is also equivariant, and thus so is . As is also a bounded self-adjoint operator, we can apply the spectral theorem to it. Observe that any closed invariant subspace of is -invariant, and is thus either or . By the spectral theorem, this forces to be a constant multiple of the identity. Similarly for . This forces to either be zero or a constant multiple of a unitary map, and the claim follows. (Thanks to Frederick Goodman for this proof.)

Schur’s lemma has many foundational applications in the subject. For instance, we have the following generalisation of the well-known fact that eigenvectors of a unitary operator with distinct eigenvalues are necessarily orthogonal:

Corollary 3 Let and be two irreducible subrepresentations of a unitary representation . Then one either has or .

Proof: Apply Schur’s lemma to the orthogonal projection from to .

Another application shows that finite-dimensional linear representations can be canonically identified (up to constants) with finite-dimensional unitary representations:

Corollary 4 Let be a linear representation on a finite-dimensional space . Then there exists a Hermitian inner product on that makes this representation unitary. Furthermore, if is irreducible, then this inner product is unique up to constants.

Proof: To show existence of the Hermitian inner product that unitarises , take an arbitrary Hermitian inner product and then form the average

(this is the “Weyl averaging trick”, which crucially exploits compactness of ). Then one easily checks (using the fact that is finite dimensional and thus locally compact) that is also Hermitian, and that is unitary with respect to this inner product, as desired. (This part of the argument does not use finite dimensionality.)

To show uniqueness up to constants, assume that one has two such inner products , on , and apply Schur’s lemma to the identity map between the two Hilbert spaces and . (Here, finite dimensionality is used to establish

A third application of Schur’s lemma allows us to express the trace of a linear operator as an average:

Corollary 5 Let be an irreducible unitary representation on a non-trivial finite-dimensional space , and let be a linear transformation. Then

where is the identity operator.

Proof: The right-hand side is equivariant, and hence by Schur’s lemma is a multiple of the identity. Taking traces, we see that the right-hand side also has the same trace as . The claim follows.

Let us now consider the irreducible subrepresentations of the left-regular representation . From Corollary 3, we know that those subrepresentations coming from different isomorphism classes in are orthogonal, so we now focus attention on those subrepresentations coming from a single class . Define the -isotypic component of the regular representation to be the finite-dimensional subspace of spanned by the functions of the form

where are arbitrary vectors in . This is clearly a left-invariant subspace of (in fact, it is bi-invariant, a point which we will return to later), and thus induces a subrepresentation of the left-regular representation. In fact, it captures precisely all the subrepresentations of the left-regular representation that are isomorphic to :

Proposition 6 Let . Then every irreducible subrepresentation of the left-regular representation that is isomorphic to is a subrepresentation of . Conversely, is isomorphic to the direct sum of copies of . (In particular, has dimension ).

Proof: Let be a subrepresentation of the left-regular representation that is isomorphic to . Thus, we have an equivariant isometry whose image is ; it has an adjoint .

Let and . The convolution

can be re-arranged as

where

In particular, we see that for every . Letting be a sequence (or net) of approximations to the identity, we conclude that as well, and so , which is the first claim.

To prove the converse claim, write , and let be an orthonormal basis for . Observe that we may then decompose as the direct sum of the spaces

for . The claim follows.

From Corollary 3, the -isotypic components for are pairwise orthogonal, and so we can form the direct sum , which is an invariant subspace of that contains all the finite-dimensional irreducible subrepresentations (and hence also all the finite-dimensional representations, period). The essence of the Peter-Weyl theorem is then the assertion that this direct sum in fact occupies all of :

Proposition 7 We have .

Proof: Suppose this is not the case. Taking orthogonal complements, we conclude that there exists a non-trivial which is orthogonal to all , and is in particular orthogonal to all finite-dimensional subrepresentations of .

Now let be an arbitrary self-adjoint kernel, thus for all . The convolution operator is then a self-adjoint Hilbert-Schmidt operator and is thus compact. (Here, we have crucially used the compactness of .) By the spectral theorem, the cokernel of this operator then splits as the direct sum of finite-dimensional eigenspaces. As is equivariant, all these eigenspaces are invariant, and thus orthogonal to ; thus must lie in the kernel of , and thus vanishes for all self-adjoint . Using a sequence (or net) of approximations to the identity, we conclude that vanishes also, a contradiction.

Given , the space of linear transformations from to is a finite-dimensional Hilbert space, with the Hilbert-Schmidt inner product ; it has a unitary action of as defined by . For any , the function can be easily seen to lie in , giving rise to a map . It is easy to see that this map is equivariant.

Proposition 8 For each , the map is unitary.

Proof: As and are finite-dimensional spaces with the same dimension , it suffices to show that this map is an isometry, thus we need to show that

for all . By bilinearity, we may reduce to the case when are rank one operators

for some , where is the dual vector to , and similarly for . Then we have

and

The latter expression can be rewritten as

Applying Fubini’s theorem, followed by Corollary 5, this simplifies to

which simplifies to , and the claim follows.

As a corollary of the above proposition, the orthogonal projection of a function to can be expressed as

We call

the Fourier coefficient of at , thus the projection of to is the function

which has an norm of . From the Peter-Weyl theorem we thus obtain the Fourier inversion formula

and the Plancherel identity

We can write these identities more compactly as an isomorphism

where the dilation of a Hilbert space is formed by using the inner product . This is an isomorphism not only of Hilbert spaces, but of the left-action of . Indeed, it is an isomorphism of the bi-action of on both the left and right of both and , defined by

and

It is easy to see that each of the are irreducible with respect to the action. Indeed, first observe from Proposition 8 that is surjective, and thus must span all of . Thus, any bi-invariant subspace of must also be invariant with respect to left and right multiplication by arbitrary elements of , and in particular by rank one operators; from this one easily sees that there are no non-trivial bi-invariant subspaces. Thus we can view the Peter-Weyl theorem as also describing the irreducible decomposition of into -irreducible components.

Remark 1 In view of (1), it is natural to view as being the “spectrum” of , with each “frequency” occuring with “multiplicity” .

In the abelian case, any eigenspace of one unitary operator is automatically an invariant subspace of all other , which quickly implies (from the spectral theorem) that all irreducible finite-dimensional unitary representations must be one-dimensional, in which case we see that the above formulae collapse to the usual Fourier inversion and Plancherel theorems for compact abelian groups.

In the case of a finite group , we can take dimensions in (1) to obtain the identity

In the finite abelian case, we see in particular that and have the same cardinality.

Direct computation also shows other basic Fourier identities, such as the convolution identity

for , thus partially diagonalising convolution into multiplication of linear operators on finite-dimensional vector spaces . (Of course, one cannot expect complete diagonalisation in the non-abelian case, since convolution would then also be non-abelian, whereas diagonalised operators must always commute with each other.)

Call a function a class function if it is conjugation-invariant, thus for all . It is easy to see that this is equivalent to each of the Fourier coefficients also being conjugation-invariant: . By Lemma 5, this is in turn equivalent to being equal to a multiple of the identity:

thus the form an orthonormal basis for the space of class functions. Analogously to (1), we have

(In particular, in the case of finite groups , has the same cardinality as the space of conjugacy classes of .)

Characters are a fundamentally important tool in analysing finite-dimensional representations of that are not necessarily irreducible; indeed, if decomposes into irreducibles as , then the character then similarly splits as

and so the multiplicities of each component in can be given by the formula

In particular, these multiplicities are unique: all decompositions of into irreducibles have the same multiplicities.

Remark 2 Representation theory becomes much more complicated once one leaves the compact case; convolution operators are no longer compact, and can now admit continuous spectrum in addition to pure point spectrum. Furthermore, even when one has pure point spectrum, the eigenspaces can now be infinite dimensional. Thus, one must now grapple with infinite-dimensional irreducible representations, as well as continuous combinations of representations that cannot be readily resolved into irreducible components. Nevertheless, in the important case of locally compact groups, it is still the case that there are “enough” irreducible unitary representations to recover a significant portion of the above theory. The fundamental theorem here is the Gelfand-Raikov theorem, which asserts that given any non-trivial group element in a locally compact group, there exists a irreducible unitary representation (possibly infinite-dimensional) on which acts non-trivially. Very roughly speaking, this theorem is first proven by observing that acts non-trivially on the regular representation, which (by the Gelfand-Naimark-Segal (GNS) construction) gives a state on the *-algebra of measures on that distinguishes the Dirac mass at from the Dirac mass from the origin. Applying the Krein-Milman theorem, one then finds an extreme state with this property; applying the GNS construction, one then obtains the desired irreducible representation.

18 comments

I do not understand why one ever states Peter-Weyl as an isomorphism of G-reps, rather than GxG-reps, where the isotypic components are actually irreducible. (That’s a good place to stop, though I suppose one could move on to the wreath product.) And yet I almost never see it presented as a GxG-theorem.

Yes, I remember you emphasising this point to me way back in grad school :-). I think that in most applications, one either needs only the G-rep version, or else one may as well go all the way to the nonabelian Fourier inversion formula. The fact that the xi-isotypic components are GxG irreducible is nice, but I don’t know of many places where this fact is actually used very much.

One step back from Peter-Weyl are the non-vanishing Schur Orthogonality Relations. The irreducible GxG-type $V\otimes V^*$ is the matrix coefficient span for V, and the tensor is unitary for GxG if the G-rep is unitary. We have two GxG-invariant inner products: one on tensors, and one on matrix coefficients.

From there, the SORs follow from Schur’s Lemma for GxG. We can think of biinvariant integration as an intertwinor of unique trivial GxG-types.

This scenario can be posed without compactness, square integrability, or even unitarity. The question becomes: what takes the place of invariant integration in the SORs? Midorikawa answers this for some classes of tempered unitary reps on semisimple Lie groups, but the general case is wide open.

I don’t understand why you portray the nonabelian Fourier inversion formula as being further than the GxG-version, insofar as it still has a “dim(V)” factor in it.

There are other statements of the form “ acts on X, such that Fun(X) is a sum of , where neither the nor repeat. For example, acting on . I don’t know a uniform statement that would include Peter-Weyl, but wouldn’t be surprised if there is one.

Well, the dim(V) factor is now only in the normalisation of the inner product, rather than in the Cartesian exponent.

Here’s roughly how I see the three facts:

G-rep Peter-Weyl: as -vector spaces, with no canonical identification.

Fourier inversion: as –Hilbert spaces, with identification given by the Fourier transform.

-rep Peter-Weyl: as -vector spaces. The -vector spaces are irreducible.

Note that the Fourier map from to (or ) is already -equivariant, so the Fourier transform is providing the canonical decomposition of into -isotypic components.

As I see it, the main thing that the -rep version of Peter-Weyl brings to the table beyond the Fourier inversion formula is the -irreducibility of the components, which in the vector space category are and in the Hilbert space category are . This irreducibility can be deduced from the inversion formula (as I do above) but I don’t know of many situations where one would need to use it, and for which one could not easily substitute the inversion formula in its place instead.

One thing that the version adds (that I also learned from Allen in grad school!) is that it encodes simultaneously the harmonic analysis for all homogeneous spaces for : for we can describe functions on (by taking -invariants on one side in Peter-Weyl) as the sum of irreps of with multiplicity spaces given by the -invariants on their duals.

This might be even more useful in the complexified form of Peter-Weyl (as decomposing regular functions on the complex group ): a great example of this is to take the maximal nilpotent , which by the theorem of highest weight has a unique invariant (up to scale) on any irrep, so we recover that is a “model space” for , with functions given by the sum of all irreps with multiplicity one, labeled by highest weights (the torus action on this line of invariants) – this is a restatement of Borel-Weil.

Here is a neat example I just ran across where the version gets you something that the -version doesn’t.

Suppose we want to compute the deRham cohomology of a compact Lie group.

If we only keep track of one -action, the -forms are . It is not obvious that this has any cohomology at all: The dimensions are completely consistent with the complex being exact. For example, if we are looking at , and we label a representation by the generating function of its highest weight vectors, then we are dealing with

.

Now, suppose we keep track of the action. Then it turns out that the multiplicity of in is . For example, when you look at , you get

.

Now it is obvious that we get cohomology in degrees and : There is no place for the trivial terms to go! And, if you recall that the -action on is trivial, it is obvious that all the other terms are exact.

[…] (for instance, one can start analysing them using the Peter-Weyl theorem, as discussed in this previous post). The global behaviour however remains more complicated, in part because the compact open subgroup […]

[…] Theorem 5 and Theorem 6 proceed by some elementary combinatorial analysis, together with the use of Haar measure (to build convolutions, and thence to build “smooth” bump functions with which to create a metric, in a variant of the analysis used to prove the Birkhoff-Kakutani theorem); Theorem 5 also requires Peter-Weyl theorem (to dispose of certain compact subgroups that arise en route to the reduction to the NSS case), which was discussed previously on this blog. […]

[…] sharpest form of the theorem, as it only describes the left -action and not the right -action; see this previous blog post for a precise statement and proof of the Peter-Weyl theorem in its strongest form. This form is of […]

[…] (which is based on the Peter-Weyl theorem combined with Schur’s lemma, and is developed in this blog post); we leave this as an exercise for the interested reader. Exercise 10 Let be subsets of a […]

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